Yes, your interpretation and ruling was spot on. Not only is Blink a VERY useful defensive tool for mages (you know how squishy they can be sometimes) as it COMPLETELY removes them from harm and detection save for things that can see into and effect the Ethereal Plane, Anything that might do Force Damage where the caster is standing, as well as anything that might happen to the caster during his or her brief stay on the Prime Material in between blinking; but it also does NOT require concentration, allowing that same spell caster to use its concentration for other nasty things in between blinking out of existence.

Most of my players do not believe that the spell would allow a character, should they roll 11 or higher at the end of their turn, to avoid attacks, spells, and other hazards when their character is not acting.

But that is PRECISELY what this spell is meant for and supposed to do. That's why it costs a resource (Spell Slot) to cast, and requires a good roll to actually work and remove the caster from combat until they appear again next turn.

If they need a realistic description of the spell? A ROUND (everything taking 1 turn in the initiative) happens in the span of 6 seconds. So every 6 seconds our dear mage who cast Blink is appearing for a brief moment to do something, then vanishing again to the Ethereal Plane. Every 6 seconds; thus appearing to be blinking in and out of existence upon the battlefield and giving us the namesake of the Spell itself.

A few counters to this spell include:

• Counterspell it as it is being cast (Perhaps one of the obvious ones)
• Dispel magic
• Etherealness
• Any area protected by Forbiddance
• Forcecage
• Magic Circle
• Mordenkainen's Faithful Hound (Can see into the Ethereal Plane)

And that was just half of the spell list I browsed. This doesn't even touch on potential creature abilities that could nullify the effectiveness of Blink, or a creature waiting in the Prime Material plane for the mage to blink back into existence for the duration of it's turn and ready an action just for that very instance. So it is certainly not a fool proof guarantee of safety, even when the dice are in Blink's favor.

Yes, you'll be free to move but must return to within 10 feet of where you entered

From the description of the Ready action in the PHB or the Basic Rules:

First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.

Moving to the Ethereal Plane is certainly perceivable, so it works as a trigger. As per this QA, there is nothing about being in the ethereal plane that prevents you from taking a reaction.

As for limitations on movement, the DMG has this to offer on the Ethereal Plane:

A traveler on the Ethereal Plane is invisible and utterly silent to someone on the overlapped plane, and solid objects on the overlapped plane don't hamper the movement of a creature in the Border Ethereal. The exceptions are certain magical effects (including anything made of magical force) and living beings. This makes the Ethereal Plane ideal for reconnaissance, spying on opponents, and moving around without being detected. The Ethereal Plane also disobeys the laws of gravity; a creature there can move up and down as easily as walking.

So you can move up to your speed, you can move through any solid objects, and you can even fly, but you can't move through other creatures (or force effects). Unfortunately, Blink stipulates:

At the start of you next turn, and when the spell ends if you are on the Ethereal Plane, you return to an unoccupied space of your choice that you can see within 10 feet of the space you vanished from.

So you can spy through walls with Blink, but you can't travel with it.